Problem: Multiply the following complex numbers, marked as blue dots on the graph: $(4 e^{23\pi i / 12}) \cdot ( e^{\pi i})$ (Your current answer will be plotted in orange.)
Solution: Multiplying complex numbers in polar forms can be done by multiplying the lengths and adding the angles. The first number ( $4 e^{23\pi i / 12}$ ) has angle $\frac{23}{12}\pi$ and radius $4$ The second number ( $ e^{\pi i}$ ) has angle $\pi$ and radius $1$ The radius of the result will be $4 \cdot 1$ , which is $4$ The sum of the angles is $\frac{23}{12}\pi + \pi = \frac{35}{12}\pi$ The angle $\frac{35}{12}\pi$ is more than $2 \pi$ . A complex number goes a full circle if its angle is increased by $2 \pi$ , so it goes back to itself. Because of that, angles of complex numbers are convenient to keep between $0$ and $2 \pi$ $\frac{35}{12}\pi - 2 \pi = \frac{11}{12}\pi$ The radius of the result is $4$ and the angle of the result is $\frac{11}{12}\pi$.